Normalized defining polynomial
\( x^{13} - 2x^{12} + x^{11} + x^{10} - 6x^{9} + 3x^{8} + 7x^{7} - 4x^{6} + 4x^{4} - 4x^{3} - 4x^{2} + x + 1 \)
Invariants
Degree: | $13$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[3, 5]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-18597951501632\) \(\medspace = -\,2^{6}\cdot 67\cdot 43517\cdot 99667\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(10.49\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
| |
Galois root discriminant: | $2\cdot 67^{1/2}43517^{1/2}99667^{1/2}\approx 1078133.5579843528$ | ||
Ramified primes: | \(2\), \(67\), \(43517\), \(99667\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{-290592992213}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$
Monogenic: | Yes | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $7$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( -1 \) (order $2$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
| |
Fundamental units: | $a$, $5a^{12}-13a^{11}+13a^{10}-3a^{9}-29a^{8}+34a^{7}+12a^{6}-27a^{5}+20a^{4}+7a^{3}-25a^{2}-3a+6$, $a^{12}-3a^{11}+5a^{10}-5a^{9}-a^{8}+6a^{7}-5a^{6}+5a^{4}-4a^{3}+2a$, $2a^{12}-5a^{11}+4a^{10}+2a^{9}-16a^{8}+17a^{7}+6a^{6}-16a^{5}+10a^{4}+3a^{3}-12a^{2}+a+3$, $3a^{12}-9a^{11}+11a^{10}-5a^{9}-17a^{8}+29a^{7}-4a^{6}-16a^{5}+18a^{4}-4a^{3}-14a^{2}+3a+3$, $a^{12}-2a^{11}+3a^{10}-3a^{9}-2a^{8}+2a^{7}-3a^{6}+3a^{5}+3a^{4}-a^{3}+2a^{2}-2$, $5a^{12}-14a^{11}+15a^{10}-5a^{9}-28a^{8}+38a^{7}+10a^{6}-30a^{5}+22a^{4}+4a^{3}-27a^{2}-a+8$ | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
Regulator: | \( 20.8860106706 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{5}\cdot 20.8860106706 \cdot 1}{2\cdot\sqrt{18597951501632}}\cr\approx \mathstrut & 0.189706526936 \end{aligned}\]
Galois group
A non-solvable group of order 6227020800 |
The 101 conjugacy class representatives for $S_{13}$ |
Character table for $S_{13}$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Degree 26 sibling: | data not computed |
Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.13.0.1}{13} }$ | ${\href{/padicField/5.7.0.1}{7} }{,}\,{\href{/padicField/5.6.0.1}{6} }$ | ${\href{/padicField/7.13.0.1}{13} }$ | ${\href{/padicField/11.9.0.1}{9} }{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.11.0.1}{11} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.5.0.1}{5} }$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.5.0.1}{5} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.11.0.1}{11} }{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{3}$ | ${\href{/padicField/37.7.0.1}{7} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.5.0.1}{5} }{,}\,{\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.10.0.1}{10} }{,}\,{\href{/padicField/43.3.0.1}{3} }$ | ${\href{/padicField/47.7.0.1}{7} }{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.7.0.1}{7} }{,}\,{\href{/padicField/53.5.0.1}{5} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.9.0.1}{9} }{,}\,{\href{/padicField/59.4.0.1}{4} }$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
$p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
---|---|---|---|---|---|---|---|
\(2\) | 2.6.6.5 | $x^{6} + 6 x^{5} + 32 x^{4} + 90 x^{3} + 199 x^{2} + 204 x + 149$ | $2$ | $3$ | $6$ | $C_6$ | $[2]^{3}$ |
2.7.0.1 | $x^{7} + x + 1$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
\(67\) | $\Q_{67}$ | $x + 65$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
67.2.1.2 | $x^{2} + 67$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
67.4.0.1 | $x^{4} + 8 x^{2} + 54 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
67.6.0.1 | $x^{6} + 63 x^{3} + 49 x^{2} + 55 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
\(43517\) | $\Q_{43517}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
\(99667\) | $\Q_{99667}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ |